PARTIAL ORDERS ON *-REGULAR RINGS

In this work we consider some new partial orders on *-regular rings. Let A be a *-regular ring, P(A) be the lattice of all projectors in A and µ be a sharp normal normalized measure on P(A). Suppose that (A, ρ) is a complete metric *-ring with respect to the rank metric ρ o n A defin ed as ρ(x,y) = µ(l(x — y)) = µ(r(x — y)), x,y (Formula presented) A, wh ere l(a), r(a) is respectively the left and right support of an element a. On A we define the following three partial orders: a (Formula presented)_s b (Formula presented) b = a + c, a ± c; a (Formula presented) b (Formula presented) l(a)b = a; a (Formula presented)_r b ^^ br(a) = a, a ± c means algebraic orthogonality, that is, ac = ca = a*c = ac* = 0. We prove that the order topologies associated with these partial orders are stronger than the topology generated by the metric ρ. We consider the restrictions of these partial orders on the subsets of projectors, unitary operators and partial isometries of *-regular algebra A. In particular, we show that these three orders coincide with the usual order < on the lattice of the projectors of *-regular algebra.